Mixing rank-one actions of locally compact Abelian groups

Using techniques related to the (C,F)-actions we construct explicitly mixing rank-one (by cubes) actions T of G=Rd₁×Zd₂ for any pair of non-negative integers d₁, d₂. It is also shown that h(Tg)=0 for each g∈G.     

Canonical extension of actions of locally compact quantum groups

In this paper, we generalize the notion of the canonical extension of automorphisms of von Neumann algebras to the case of actions of locally compact quantum groups (in the sense of Kustermans and Vaes). Various expected properties will be shown to hold for this new canonical extension. As an application, we describe the flow of weights of the crossed product of a type III factor by some special action of a discrete Kac algebra.     

Products of locally compact groups with zero and their actions

In this paper we briefly describe the structure of a product of locally compact groups with zero, and then investigate actions of such semigroups on Hausdorff spaces. Communicated by A. D. Wallace     

On locally finite minimal non-solvable groups

In the present work we consider infinite locally finite minimal non-solvable groups, and give certain characterizations. We also define generalizations of the centralizer to establish a result relevant to infinite locally finite minimal non-solvable groups.     

Universal G-spaces for proper actions of locally compact groups

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w(G)?τ, can act properly on Rτ?{0} such that Rτ?{0} contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ?τ. The metric cones Cone(G/H) with H⊂G a compact subgroup such that G/H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H⊂G does not exceed the weight of G.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

P-amenable locally compact groups

LetG denote a locally compact group andP(G) [P _c (G)] the topological semigroup of absolutely continuous [absolutely continuous and compactly supported] probability measures onG. We say thatG isP-amenable [P _c -amenable] if the topological semigroupP(G)[P _c (G)] is amenable. Some combinatorial properties of this class of groups are studied. The relationship between amenability andP-amenability ofG is investigated. It is shown that for a connected solvableP _c -amenable groupG, G has polynomial growth if, and only if,P _c (S)={fP _c (G)suppfS} is amenable for any open subsemigroupS ofG.This paper contains a part of the author's doctoral thesis at the State University of New York at Albany. The author wishes to thank ProfessorJoe Jenkins for his valuable suggestions and encouragement during the course of this work.     

On spectral analysis in locally compact motion groups

Beurling's theorem in spectral analysis of bounded functions on the real line is generalized to a class of locally compact motion groups and to Heisenberg groups.     

On contraction groups of automorphisms of totally disconnected locally compact groups

We prove that recent results of Baumgartner and Willis on contraction groups of automorphisms of metrizable totally disconnected locally compact groups (Israel J. Math. 142 (2004), 221–248) remain true for non-metrizable groups. Supported by an NSERC Grant.